Equivalence classes of homotopy-associative comultiplications of finite complexes

JOURNAL OF PURE AND APPLIED ALGEBRA ELStiIER

Journal of Pure and Applied Algebra 102 (1995) 109-136

Equivalence classes of homotopy-associative comultiplications of finite complexes Martin Arkowitz”, Gregory Luptonbv’ ‘Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA bDepartment of Mathematics, Cleveland State University, Cleveland, OH 441 IS, USA

Communicated by J.D. Stasheff; received 1 October, 1992;revised 22 September 1993

Abstract Let X be a finite, l-connected CW-complex which admits a homotopy-associative comultiplication. Then X has the rational homology of a wedge of spheres, S”’ ’ ’ V . . . V Snr’+‘. Two comultiplications of X are equivalent if there is a self-homotopy equivalence of X which carries one to the other. Let G?‘,(X), respectively g,,(X), denote the set of equivalence classes of homotopy classes of homotopy-associative, respectively, homotopy-associative and homotopycommutative, comultiplications of X. We prove the following basic finiteness result: Theorem 6.1 (1) If for each i, (a) ni # nj + nk for every], k withj < k and (b) ni # 2nj for every j with nj even, then g,(X) is finite. (2) g,,(X) is always finite. The methods of proof are algebraic and consist of a detailed examination of comultiplications of the free Lie algebra it++(DX) @JQ. These algebraic methods and results appear to be of interest in their own right. For example, they provide dual versions of well-known results about Hopf algebras. In an appendix we show the group of self-homotopy equivalences that induce the identity on all homology groups is finitely generated.

1. Introduction

In this paper we consider various sets of comultiplications of a topological space, up to a suitable notion of equivalence. A comultiplication of a space X is a map cc:X + X V X such that pa and p’a are both homotopic to the identity map of X, where p and p’ are the projections X V X + X onto the first and second summands of the wedge. Examples of comultiplications abound, and a large and important class occurs when X is a suspension and a is the natural pinching map. Two comultiplications a and /3 of a space X are equivalent if there is a homotopy equivalencef:X + X such that fif and (f V f )a:X + X V X are homotopic. There are natural definitions for a comultiplication a to be homotopy-associative or homotopy-commutative

1The second author’s work was supported in part by a Cleveland State University Research and Creative Activities award. 0022-4049/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved. SSDI 0022-4049(94)00074-S

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which we recall below. These properties are preserved under equivalence and we investigate equivalence classes of comultiplications with these properties. In particular, we consider basic finiteness questions such as the following: For which spaces are there finitely many equivalence classes of homotopy-associative comultiplications? For which spaces are there finitely many equivalence classes of homotopy-associative and homotopy-commutative comultiplications? We give rather complete answers to these questions. In the dual context, multiplications of a space X have been extensively studied, and it is known that the possibilities are limited. Curjel has shown that any finite complex admits finitely many equivalence classes of homotopy-associative multiplications [9, Theorem I]. The questions concerning comultiplications that we examine here are dual to those in [9]. In general, however, the results for equivalence classes of comultiplications of finite complexes are more diverse than for equivalence classes of multiplications of finite complexes. For instance, S3 V S5 admits infinitely many equivalence classes of homotopy-associative comultiplications (Example 6.6( 1)). Let %?(X)denote the set of homotopy classes of comultiplications of X, and q=(X), respectively QZa,(X),the set of homotopy classes of homotopy-associative comultiplications, respectively of both homotopy-associative and homotopy-commutative comultiplications, of X. We denote the equivalence classes by g(X), G?:,(X)and g,,(X). Our main results concern a finite complex X which admits a homotopy-associative comultiplication. It is known that X and a wedge of spheres S”‘+ ’ V ... V Snr+ ’ have the same rational homotopy type. We prove in Theorem 6.1 that if for each i, (a) ni # nj + nk for every j, k with j < k and (b) ni # 2nj for every j with nj even, then g,(X) is finite. Also in Theorem 6.1, we prove that U,,(X) is always finite. The paper is organized as follows: In this section we give definitions and fix notation. In Section 2 we establish basic facts concerning comultiplications and rationalization. The main result is Proposition 2.3 which enables us to obtain information about comultiplications of a space X by studying comultiplications of the rational homotopy Lie algebra rr# (QX) @ Q of X. The latter is a purely algebraic and very effective context in which to work. In Section 3 we relate G?(X) to equivalence classes of comultiplications of the corresponding rational homotopy Lie algebra rc# (0X) 0 Q. The main result here is Theorem 3.14, which essentially allows us to conclude that G?(X), g,(X) or g,,(X) are finite whenever the corresponding equivalence classes of Lie algebra comultiplications of n,(&?X)@ Q are finite. In Sections 4 and 5 we work in an algebraic setting and show that certain Lie algebras admit only finitely many equivalence classes of comultiplications. These results are applied in Section 6, but some of them are interesting in their own right. Theorem 4.4 asserts that an associative comultiplication on a Lie algebra is determined up to equivalence by its quadratic part. Proposition 5.3 and Corollary 5.5 are dual to well-known results about Hopf algebras. The main result, Theorem 6.1, then follows easily from the previous sections. Also in Section 6 we give several examples that illustrate our results. In an appendix we prove that, for a finite complex, the group of self-homotopy equivalences that induce the identity on homology groups is finitely generated. Our

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work uses this result, but it also seems of independent interest and is not available in the literature. We end this section with conventions and notations that are adopted and used throughout the paper. A topological space will either be a l-connected, based space of the homotopy type of a based CW complex or the rationalization of such a space. All maps and homotopies are to preserve basepoints. We do not distinguish notationally between a map and its homotopy class. However, we sometimes signify homotopy of maps by N and same homotopy type of spaces by z . For spaces X and Y, we let [X, Y] denote the set of homotopy classes of maps from X to Y. If X is a space, then the graded homotopy group of X is denoted by nj# (X). We also consider graded Lie algebras over the rationals Q, referred to as Lie algebras, and homomorphisms of Lie algebras. The rational homotopy Lie algebra of a space X is x# (G?X)0 Q, where s2X is the loop-space of X and the bracket operation is obtained from the Samelson product [23, Ch.X, Section 5). We often denote this Lie algebra by Lx. IffiX-+ Y, thenf#:Lr = 7r#(s2X)@Q +Ly = n#@Y)@Q denotes the induced homomorphism of Lie algebras. A Lie algebra L isfree if there is a graded Q-vector space V with L r [L(V), the free Lie algebra generated by V (see [21, p. 161). All Lie algebras in this paper are free Lie algebras. In addition, we only consider the case where V is positively graded and finite-dimensional, so that each V, is a finitedimensional vector space with V, = 0 for n I 0 and for n sufficiently large. If i, . . . ,y,> is a graded basis for V, then we write R(V) = O_(y,, . . . ,y,.), and i:: 1, ... 9yI} is called a basis or a set ofgenerators for the Lie algebra [L(V ). We say x E L(y,, . . . ,Y,) ifx E L(Y,, . . . , y,.). for some n, and write 1x 1for the degree n of x. If XE [L(y,, . . . , y,),then x has length k if x can be expressed as a linear combination of brackets of length k in the generators y,, . . . ,y,. A Hall basis for U_(V)is a totally ordered, graded vector space basis for the underlying graded vector space of k(V). Such a basis is constructed as follows: Choose a totally ordered basis y, < ... < y, for T/.Then the remaining basis elements for [L(I/ ) are certain monomials in the y;s, using the bracket operation as product, which are chosen and ordered by proceeding inductively over bracket length. Once the elements of length I s have been chosen and ordered, this determines the basis elements of length s + 1 according to certain rules (see [l l] or [20, LA, Section 4.5)]). The length s + 1 basis elements are then ordered arbitrarily amongst themselves, and each is given greater order than any element of length I s. We remark that in the graded case, a Hall basis for [L(V) includes not only the ‘basic products’ of [ 111, but also the squares of basic products of odd degree (see [18, 4.53). The coproduct of Lie algebras L and L’ is denoted L u L’ and defined as follows: If L = [L(V) and L’ = [L(V’), then L u L’ = [L(V @ V ‘). A (Lie algebra) comultiplication of L is a homomorphism 4: L -+ L u L such that rcb and ~‘4 both equal the identity homomorphism of L, where n and 7~’are the projections L u L AL onto the first and second summands (see [4, Section 23). Elements of L U L in either summand are distinguished from each other by using a prime on elements from the second summand. Thus, if L = lL(y,, . . . ,y,), then LU L = [L(y,, . . . ,y,,y;, . . . ,y:).

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Some of our proofs in Section 3 use the universal enveloping algebra functor U from the category of Lie algebras to the category of associative algebras (see [19, Appendix B] or [21] for details). In particular, a Lie algebra L(V) has universal enveloping algebra U([L(Y )) = T (V ), the tensor algebra on I/, with inclusion i: U_(V) + T(V) the unique Lie algebra homomorphism that extends the identity on V. The coproduct of tensor algebras T (V ) and T (V ‘) is denoted T (V ) u T (V ‘) and defined by T(V) u T(V’) = T(V 0 V’). A comultiplication of T(V) is a homomorphism $ : T(V) + T( V ) u T(V) such that nlC/ and n’$ both equal the identity homomorphism of T(V), where 71and z’ are the projections T(V) u T(V) + T(V) onto the first and second summands (see [7]). By applying the universal enveloping algebra functor to a Lie algebra comultiplication 4: IL(V) + I_(V) U I_(V), one obtains a comultiplication U(4): T(V) + T(V) U T(V). If 01:X + X V X is a comultiplication of X, then the pair (X, a) is called a co-Hspace. Often we omit reference to the comultiplication and call the space X a co-Hspace. If X is a co-H-space which is a finite CW complex, then it is known [6, Theorem 2.21 that X has the same rational homotopy type as a wedge of spheres sn,+1 V . . . V yr+l, n1 < . . . I n,. It follows that the rational homotopy Lie algebra Lx = rc#(sZX)@Q is a free Lie algebra lL(y,, . . . ,y,) with lyil = ni. The sequence of integers (nt, . . . ,n,) is called the type of the co-H-space X. Note that the type is determined either by Lx or by the rational homology H,(X; Q). A comultiplication ct:X+XVX which satisfies (1 Vcr)az(aVl)a:X+XVXVX is called homotopy-associative and the co-H-space X is also called homotopy-associative. By [ 13, Theorem 2.3) a homotopy-associative co-H-space always has homotopy inverses and is therefore a cogroup. Thus the term cogroup will mean homotopy-associative co-H-space. A finite cogroup is a cogroup which is a finite CW complex. A prime example of a finite cogroup is the suspension of a finite complex with the natural pinching map as comultiplication. A comultiplication a: X -B X V X is called where T: XVX+XVX is the homotopy-commutative if Tcr N a: X +XVX, switching map which interchanges coordinates. Many of these terms carry over to a multiplication 4 of a Lie algebra L. We call 4 associative if (1 U +)# =(~~l)~:L-,L~L~Landwecall~commutatioeifT~=~:L-,L~L,where T : L U L + L I-IL is the switching homomorphism which interchanges summands.

Let b(X) denote the group of homotopy classes of self-homotopy equivalences of a space X and d,(X) the subgroup of B(X) of homotopy equivalences that induce the identity on all homology groups. Let Aut L denote the group of automorphisms of a Lie algebra L and Aut,L the subgroup of automorphisms that induce the identity on the vector space of indecomposables L/CL, L]. The group 8’(X) acts on g(X) as follows: If f~ b(X) and CIE w(X), then f* a is the composition

which is a comultiplication of X. The set of orbits under this action are the equivalence classes 4(X) of comultiplications. Clearly homotopy-associativity and

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homotopy-commutativity are preserved by this action and so 8’(X) also acts on U,(X) and %JB,,(X).The corresponding sets of orbits are the sets of equivalence classes g,(X) and g,,(X). Likewise, for a Lie algebra L, Aut L acts on W(L), the set of comultiplications on L: If 0 E Aut L and 4 E g(L), then 0* 4 = (0 u 0)@-‘. Writing the set of associative comultiplications on L as wa(L) and the set of associative and commutative comultiplications as GJT~,(L), we have the corresponding sets of equivalence classes denoted G??:,(L) and ga,(L). In addition, we often consider the action of the subgroup b,(X) on the sets w(X), 9Za(X) and wa,(X), and the subgroup Aut,L on the sets W(L), Va(L) and %Ta,(L).Our reasons for doing so are explained in Remark 3.15. In these cases we use the notation S//G to denote the set of orbits or equivalence classes under the action of a group G on a set S. We call a function f: A -+ B of sets jinite-to-one if the inverse image of every point of B is a finite set. For a Lie algebra L, W(L) and Aut, L are conveniently described in terms of a given If L = [L(y,, . . . , yV), then as above L u L = set of generators. L(Y,, **f,y,,y;, *a*, y:). A homomorphism 4: L + L u L is a comultiplication if and only if, for each Yj, d(Yj) = Yj + Y>+ C Ps(Yj)7

(1.1)

sz2

where Ps(Yj) is a polynomial of length s in y,, . . . , y,, y;, . . . , y:, each monomial of which contains at least one entry from yi, . . . ,y, and at least one entry from y;, *.* , y:. Here, and in what follows, polynomials in a Lie algebra are obtained by as multiplication. We call P, defined by using the bracket operation Ps(yj), the perturbation of the comultiplication &‘4, p. 843. Similarly, an p(Yj) = C, 2 2 automorphism 19:L + L is in Aut,L if and only if, for each generator Yj, @Yj) = Yj + C Qs(Yjh

(1.2)

s>2

where Q,(Yj) is a polynomial of length s in y,, . . . ,y,.

2. Rationalization and comultiplications We review some relevant facts concerning rationalization or &P-localization of groups and spaces; for more details, see [12]. A group G is called Q-local if for every positive integer n, the map x -+ x” is a bijection G -P G. A homomorphism f : G -P H of groups is called a Q-isomorphism if (1) the kernel off is a torsion group and (2) for every x E H, there is a positive integer n such that x” is in the image of J: Next, a topological space Y is called a rational space if Zi( Y) is &P-localfor every i. For any space X, there exists the rationalization of X, written Xo, which is a rational space, and the rationalization map 1: X -+ Xo which is universal with respect to maps of X into rational spaces. Spaces X and Y are said to have the same rational homotopy type if Xo = Y,. The rationalization construction is functorial so that y E [X, Y ]

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yields y. E [Xo, Yo]. Thus we have a rationalizationfunction e: [X, Y ] --P[Xc, Yo], defined by e(r) = yo. One easily sees that the rationalization function restricts to functions e: q(X) + %3(X,), e: go(X) + G9JXo), e: 8(X) -+ 8(X,), e: b,(X) + 8,(X,) and so on. We now recall some rational homotopy theory; for details see [21]. A basic result asserts that each rational space X has a Quillen model, i.e., a differential graded Lie algebra minimal model L(X). Furthermore, there is a notion of homotopy for homomorphisms between Quillen models. The Quillen model is functorial and for rational spaces X and Y it gives a bijection between homotopy classes [X, Y ] and (differential graded Lie algebra) homotopy classes [L(X), L(Y )]. Next, consider a co-H-space X. Since the rational homotopy Lie algebra Lx = n++(f2X) 0 0 is a free Lie algebra, it follows that the Quillen model of Xo is just Lx with zero differential. But if X is a co-H-space, then so is X V X, and the Quillen model of (X V X), is again just the rational homotopy Lie algebra Lxvx = n,(s2(X V X))@Q = Lx u Lx with zero differential. Now for homomorphisms between Quillen models that have zero differentials, homotopy reduces to equality. So for co-H-spaces X and Y, CL(Xo), W’dl = Horn(Lx,-&I. Hence the set of Lie algebra homomorphisms Hom(Lx, Lx u Lx) is in one-one correspondence with the set of homotopy classes of maps [Xo,(X V X),]. Moreover, this correspondence restricts to give bijections between W(L,) and 9?(X,),%JL,) and %?JX,), and V,,(L,) and %‘O,(X,) (see [4, Section 21). In a similar way, if X is a co-H-space, then we identify Aut Lx with 8(X,) and Aut, Lx with 8,(X,). If a E w(X), then the induced homomorphism c(# : nn(s2X)@Q -+ n,(s2(X V X))@Q is a Lie algebra comultiplication of Lx. Thus there is a function g: %7(X)+ %?(L,) defined by g(a) = CI#. By the above discussion, this function can be identified with the restriction of the rationalization function e: G!?(X)+ %(Xo). Similarly, an element f~ 8(X) induces f# E Aut Lx, and if f~ S,(X), then f# E Aut, Lx. This gives a homomorphism h: 6’(X) + Aut Lx that restricts to h: b,(X) -+ Aut, Lx. Again, by the above discussion, we identify this function with the restriction of the rationalization function e: 8, (X) + 8, (Xo). We begin by interpreting %?(X)in terms of homotopy sets. If X is a cogroup with comultiplication a, then it is well-known that c1induces a group structure - denoted multiplicatively - on the set [X, Y 1, for any space Y. Now let X bX be the space of paths in X x X which begin in X V X and end at the basepoint of X xX and let j: X bX + X V X be the map that projects a path onto its initial point. In other words, X b X is the homotopy-fibre of the inclusion X V X + X x X. Let X be aJinite cogroup. Thefinction Y: [X,X bX] + W(X) dejined by !P(B) = CI.(ja) is a bijection, where c1is the given comultiplication of X.

2.1. Lemma.

We omit the proof of Lemma 2.1; the dual case is well-known (see, for example, [l, Lemma 21).

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2.2. Lemma. (cf. [12, Corollary 6.51 and [S, V, Proposition 5.33). Zf X is a jnite cogroup and Y is a space such that Zi( Y ) is finitely generated for all i, then [X, Y] is a finitely generated nilpotent group and the homomorphism 1,

: cx, y I+ cx, YQll

induced by the Q-localization map 1: Y + YQ, is a Q-isomorphism.

Proof. Let Y (II)denote the nth Postnikov section of Y and let p: CQX --t X be the map defined by p(f, t) = f(t) for f e QX and 0 5 t I 1. Since X is a cogroup, it is well-known that there is a maps: X + CQX such that s is a co-H-map, i.e., is compatible with the comultiplications on X and CQX, and such that ps = l[lO, Theorem 2.23. Thus for some n we have the commutative diagram [X,

Y]

I

N

Y(")]-2 A

I

E

[X, Yb’l

pnx, Y@)]

I

z

[‘“’ *

I’“’ *

1,

[X,Y,]

[X,

pf,

2-

[CL?X,

Y$'] z

[QX,OY'"']

I b”T] [ax, (QP’)

f2Y

with p* a one-to-one function and s* an epimorphism. One proves, as in [3, p. 161, by induction on n that [ax, QY(“)] is a finitely generated nilpotent group. Observe that nilpotency also follows from [23, X, Corollary 3.81 because [QX, SZY’“‘]g [(DX)“, QY’“‘] for some skeleton (QX)N of s2X. Since s* is an epimorphism, [X, Y ] is a finitely generated nilpotent group. To complete the proof it suffices to show that (521(“))*is a Q-isomorphism. Note that Szlt”) is just the Q-localization map 1: s2Y (“)+ QY $‘. Therefore, we need only prove that 1,: [s2X,sZY’“‘] + [.QX,OY$‘] is a Q-isomorphism. For this, one establishes by induction on n that 1,: [A, QY’“‘] + [A, QY g’] is a Q-isomorphism for all n 2 2 and all connected - but not necessarily l-connected - spaces A such that H,(A) is finitely generated for all i. 0 We now use the bijection Y of Lemma 2.1 to transfer group structure to the set W(X). 2.3. Proposition. Let X be a$nite complex. (1) a,(X) is afinitely generated group and h: b,(X)

+ Aut, Lx is a Q-isomorphism. (2) Zf X is a cogroup, then the sets %7(X) and W(Lx) can be given group structure such that ‘Z(X) isfinitely generated and g: W(X) + %(Lx) is a Q-isomorphism. Furthermore, the kernel of g is jnite.

Proof (1) As in the above discussion, we identify Aut, Lx with &‘,(Xo) and h: b,(X) + Aut, Lx with the rationalization function e: b,(X) + S,(Xo). By the Appendix, a,(X) is a finitely generated group. By [16], e: g*(X) +8,(X,) is a Qisomorphism - in fact, the rationalizing homomorphism.

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(2) We use bijections Y and Y’ as in Lemma 2.1 and note that the following diagram commutes:

’

W(X)9 I %(I,,) = %(X,)

[X,XbX] e 1

z

[X,,

where e is the rationalization [X,XbX]L

X0 bXQ]

function. Since e is the composition

CXWbX),,(Xb-QJ z C&,-GbXcJ,

where 1: X bX + (X bX), is the Q-localization map, it follows from Lemma 2.2 that e, and hence g, is a Q-isomorphism. Next, [X,X bX] is a finitely generated nilpotent group by Lemma 2.2. Since e: [X,X bX] + [XQ, XQ bXo] is a CD-isomorphism, the kernel of e is a torsion subgroup of [X,X bX]. But every torsion subgroup of a finitely generated nilpotent group is finite [(15, p. 2321 and [3, Section 21). Thus the kernel of g: q(X) + %(Lx) is finite. 0 Nowlet L = [L(yI, . . . , y,) and denote by Int lL(y,, . . . ,y,) the subset of L consisting of all elements which can be written as a polynomial in y,, . . . ,yr with integer coefficients. We remark that an element is in Int [L(y, , . . . , yr) if, and only if, when it is written as a linear combination of elements from a Hall basis constructed from {YI, ... ,Y,>, the coefficients are integers. An element 4 E W(L) is called integral (with respect to the generators y,, . . . , y,) if 4(yi) E Int lL(y,, . . . ,y,,y;, . . . , y;) for each i = 1, . . . ,r. Let Int%Z(L(yl, . . . , yl)) denote the set of integral Lie algebra comultiplications. Similarly, an element 0 E Aut L is called integral (with respect to yl, . . . , y,) if B(yi)EIntL(yl, . . . , y,) for each i. Then Int Aut L(y,, . . . , yr) denotes the set of integral automorphisms and Int Aut, L(y, , . . . , y,) denotes the integral automorphisms in Aut, L. 2.4. Lemma. If X is ajnite cogroup, then there is a set of generators x1, . . . ,x, for the Lie algebra Lx = II#(S~X)QQ such that: (1) Int%(L(xi, . . . ,x,)) is a subgroup ofV(L,) with respect to the group structure of Proposition2.3, and IntAut, lL(xi, . . . ,x,) is a subgroup of Aut, Lx; (2) if g: G%‘(X)+ %(L,) and h: J’,(X) + Aut, Lx are as defined abooe, then Imageg G Int%(L(xi, . . . ,x,)) and Imageh E IntAut, L(xl, . . . ,x,).

Proof. By Proposition2.3, there exist finite sets of generators al, . . . ,a. of the group w(X) and fi, . . . ,fb of the group b,(X). Let a0 E q(X) be the cogroup comultiplication of X and let fo:X + X be the canonical inverse map with respect to ao. We claim that there exists a basis x i, . . . ,x, of Lx such that ai# and fj# are integral with respect to this basis, i = 0, . . . ,a and j = 0, . . . ,b. By applying the claim to a0 and f. we see that IntV(IL(x,, . . . ,x,)) is a subgroup of %?(L,). Lemma 2.4 then follows. To prove the

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claim let y, , . . . ,y, be any basis of Lx. We show by induction that for every s = 1, . . . ,I, there exists a basis x1, . . . ,xs,ys+ 1, . . . ,y, of Lx such that all ai+ (xk) and fi, (x,J are integral and furthermore that xk = Nkyk for positive integers Nk, for each k = 1, . . . ,s. Let sj = - 1 for j =0 and sj= 1 for j>O. Since ai#(yl) =y, +y; and fj,(yi) = sjyi, the result holds for s = 1 with x1 = y,. NOW suppose that the inductive hypothesis holds for s. We write, for each i and j, ai#(Ys+l)=Ys+l

+Y:+I

%,I)

+C-YI, I bV,I,

where each Yr is a monomial of length 2 2 in the generators y,, . . . ,y,, y;, . . . ,y:, each Y, is a monomial of length 2 2 in the generators yl, . . . ,y,, and a(i,r), bti,l)y ,!hCC each yk that appears in Yr or Y, iS (l/Nk)Xk and each 4j.n and b&J, are integerS. y; that appears in Yr is (l/N,)x;, we have %+(Ys+i)

= Ys+l + Y:+I

fj#(Ys+l)=EjYs+l

+

7%x1,

+TeXJv

for some integers cti.1) and c;j,J), where each X1 is a monomial of length 2 2 in xi, . . . ,xs, x;, . . . ,x: and each XJ is a monomial of length 2 2 in x1, . . . ,xs. We set

Then we have ai#(Ns+lys+l)=N,+lys+l fj#(Ns+lY,+l)

= sjNs+iys+l

+N,+iy:+l

+C&,rjXr, I

+ Tdij.J,X,

for some integers da,r, and dij,n. Set x,+ 1 = N, + 1y,+ 1 to complete the induction. The claim then follows by taking s = r. q 2.5. Remarks. (1) In the rest of the paper we use the generators x1, . . . ,x~ of Lx constructed in Lemma 2.4 without explicit mention, and so the conclusion of Lemma 2.4 will hold. (2) Although Lemma 2.4 is dual to Lemma 5.2. of [9], the proof that we have given is not dual to Curjel’s proof. We have used localization methods applied to the groups b,(X) and Q?(X), which were not available earlier. We have also avoided the dual of the condition that the rational cohomology algebra be primitively generated, which was used in Curjel’s argument.

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As a consequence of Lemma2.4, the homomorphisms phisms g:%(X) + Int%7([L(x1,. . . ,x,))

and

g and h induce homomor-

h: 8,(X) + IntAut,[L(x,,

. . . ,x,).

(2.3)

The following is now an immediate consequence of Proposition 2.3. 2.6. Corollary. IfX is ajnite

cogroup, then the homomorphism g hasfinite kernel and the homomorphism h has the property that for every 0 E IntAut,[L(xl, . . . ,x,), there is a positive integer n such that 0” E Image h.

3. Reduction to Lie algebras

In this section we relate the set Q?(X)//&,(X) to the set 9?(Lx)//Aut,Lx, in such a way so as to reduce a study of the former to a study of the latter. This is done by first relating the set +?(X)//&,(X) to the set Int%(Lx)//IntAut, Lx (Proposition 3.4) and then relating the set Int%(Lx)//IntAut,L, to the set %(Lx)//Aut,Lx (Proposition 3.12). Most of the proofs in this section proceed by induction over a given set of generators for a Lie algebra L. We fix a set of generators {x1, . . . ,x,.} of k(F) with [Xii = ?li and nl I ... I n,. In the case that X is a finite cogroup, L = Lx = nn#(QX) @ Cl!and the generators are chosen as in Lemma 2.4. Let %’ denoted a Hall basis for L, constructed from xi, . . . ,x,. Let $= z - {x, . . . ,x,), so that 2 consists of the decomposable elements of 2. We write S? = {X1,X2, . ..}. 3.1. Definition. An automorphism for some m E { 1, . . . ,r}, we have

$ E Aut, L is an elementary automorphism

of L if,

for some Xj E 2, where 1x, 1= IXjl. Note that $ is in Int Aut, L. For every positive integer n, let El” be the subgroup of Int Aut, L generated by the set of nth powers of elementary automorphisms of L. 3.2. Lemma. For every n, El” has finite index in IntAut,[L(x,,

. . . ,x,).

Proof. We describe a finite set of coset representatives for IntAut, L/El”. Let ~9E Int Aut, L and assume inductively that there exists $(kj E El” satisfying, for each i I k, ($(k)‘O)(Xi) = Xi + CUjXj

A4. Arkowitz,

G. Lupton / Journal of Pure and Applied Algebra 102 (1995) 109-136

119

with a: integers such that 0 I aj I n - 1, where the sum is over allj with Xj E 2 and JXjl = [xii. Then on the next generator, (+(kj’@)(xk+1)=

Xk+i

+

&5+’

+

nb5+‘)Xj,

for integers a! + 1 and bjk+’ with 0 5 at” I n - 1. Now consider the elementary automorphisms $j defined by +j(x k+ 1) = xk+ I -I- Xj for each j in the latter sum. A straightforward calculation shows that for i I k + 1,

to complete the induction. Therefore for any 8 E IntAut,L, go IntAut,L given by B(Xi)

=

Xi

+

the coset of 8 in IntAut, L/El” contains some

CUfXj,

where 0 I aj I n - 1. Since the indexing sets forj are finite, for each i = 1, . . . ,r, there are finitely many such 8: 0 3.3. Lemma. Let X be aJinite cogroup and let h: a,(X) + IntAut, L(xl, . . . ,x,) be the homomorphism in (2.3). Then Image h has jinite index in Int Aut, L(xl, . . . ,x,). Proof. By Lemma 3.2, it suffices to show EIN E Imageh for some N. Let $5 be the elementary automorphism defined by +$(xi) = xi + Xj for Xj E 2. By Corollary2.6 there is some positive integer n(i,j) such that ($jy(iJ) E Imageh. There are finitely many elementary automorphisms I/; and we let N be the product of all the n(i,j). Therefore (I&)~ E Image h for all i, j. Hence EIN E Image h since EIN is generated by the

(lclf)“. 0 The homomorphisms induce a map P:V(X)//B,(X) 3.4 Proposition.

g: U(X) + Int%?(L,) and h: &‘.JX) + IntAut,L,,

-+ Int%(L,)//IntAut,L,.

as in (2.3),

(3.4)

The map p is finite-to-one.

Proof. For tl E VT(X), we write (a) for its equivalence class in @‘(X)//B,(X), and for

4 E Int%(&) we write [$] for its equivalence class in Int%(L,)//IntAut,L,. By Lemma3.3, there are finitely many cosets in IntAut,L,/Imageh, so let {e,, . . . ,0,}

120

M. Arkowitr,

G. Lupton /Journal of Pure and Applied Algebra IO2 (1995) 109-136

be a set of coset representatives. Then any 8 E IntAut,& can be written 8 =f# 08~ and for somej = 1, . . . ,s and some J-E&‘,(X). Now let [b] E IntV(Lx)//IntAut,Lx For each j = 1, . . . ,s, let consider the set (0, * 4 , . . . ,&*c$} c Int%(Lx). which induce tIj*4, i.e., be all comultiplications {a{, . . . ,a&,} E g(X) {ai, . . . ,a&} = g-‘(Oj*q5). B y C orollary2.6 this set is finite. We now show that be such p-‘[$I = {, ... ,,... ,,... ,(a&)>. Let (a) E %(X)//S,(X) thatp(a) = [$].Then thereexists 6’E IntAut,&witha# = O*$. But 0 =f+oOj, for somej = 1, . . . ,s and some f~ d,(X), and so a+ = (f# 0 Oj)* 4 =fx *(ej*#). Therefore (J- l *a)# = f# ’ *a# = Bj* 4. By construction, f- ’ *a = ai for some i. Thus (a) = (a{) and the proposition is proved. 0 We now relate the orbit sets Int%(Lx)//IntAut,Lx

and %?(Lx)//Aut,Lx.

3.5. Definition. Let s be an integer 2 2 and let Q(s) = {p/q E Q: q 2 2, qls! and ,x,) we write O(xi) = xi + Es 2 2Qs(xJ, as in (1.2). We OIp
8oeE9. Proof. Since sQs(xJ E Int L(xi , Ts(xi) E Int L(x~ , . . . ,x,) and Rs(xi) =

. . . ,x,)

we

write

Qs(xi) = R,(xi) + 7’s(xi), with

1 Ct.j Ys,j, j

where cf,j E G!(s) and Y,,j has length s and degree Ixil. We ‘remove’ the integral part 7’s(Xi),proceeding inductively over the generators as in Lemma 3.2. Assume inductively that there exists et E Int Aut, IL&, . . . , x,), such that (O,o @(xi) has length s part as in (3.9, for all s 2 2 and i I k. Suppose o(xk+l)=xk+l

+

1 922

(&(Xk+d

+

Ts(xk+l))

M. Arkowitz, G. Lupton JJournal of Pure and Applied Algebra 102 (1995) 109-136

121

as above. Then (~koe)(xk+l) = ektXk+d+

c

(ek~~axk+d +

sz2

c (ekw(xk+d.

(34

s>2

By assumption t!&E Int Aut,U_(xl, . . . ,x,), so Ok(xk+1) and each (fI, 0 7’J(xk+ 1) are in Int [L(xl, . . . , x,). Also, (4 o &)(xk+ 1) = Cj ct.) ’ e,( Y,,j), where c!,: ’ E Q(s). Since OkE Int Aut, [L(xl, . . ,x,), each &(Y,,j) E Int lL(x1, . . . ,x,) and is an integral linear combination of brackets of length 2 s. We collect together the terms of homogeneous length t that are contributed by the summands (O,o Rs)(xk + 1) for s I t. For each s I t, the denominators of the coefficients in R,(x k+l) divide s!. Thus the length t part of (O,o Rs)(xk+ 1) can be written as a sum of terms from Int [L(x,, . . . ,x,.) plus a sum of terms having rational coefficients whose denominators divide s! and hence t!. It follows that

C (ekoRs)(Xk+l)= C

C"f,:'yt,j

122

st2

+

1 uf+l~ 322

j

with LJi+l E Int [L(xi, . . . ,xI) of length s and u:,‘j’ E Q(t). We rewrite (3.6) as (ok”~)(xk+d=xk+l

+

1 fk2

cd,;‘yt,j+ j

1 s>_2

~s(xk+l)~

with v,(x,+l) E Int lL(Xl, . . . ,x,),anda~,~‘~Q(t).Define~~IntAut,IL(x,,...,x,)by $(xk+l)

=

xk+l

-

1

vs(xk+l),

st2

for i # k + 1. Then ($0 ok0 @(xi) = (e,@)(xi) for i = 1, . . . , k and k+ ’ Y,.j. The inductive step is completed by setting (ll/“~ko~)(xk+l)=xk+l +CtCjhgJ 8k + 1 = II/ 0 ok. Induction starts with k = 1, where we take 81 = 1. The lemma follows by setting 8= 8,. 0 and

Il/(Xi)

=

Xi

We now prove some technical results concerning the universal enveloping algebra T(V) of a Lie algebra O_(V).We use %‘(T) to denote the set of comultiplications of a tensor algebra T (see Section 1). We denote by Aut, T the automorphisms of T which induce the identity on the indecomposables. If {xl, . . . ,x,} is a basis for I’, then we write T(V) = T(xl, . . . ,x,), and with respect to this choice of generators, Int T(xl, . . . ,x,) denotes the subalgebra of T(V) consisting of all polynomials in that have integer coefficients. Then Int V(T(x,, . . . ,x,)) and Int Aut, Xl, ... 7 x, T(xl, . . . ,x,) are defined in analogy to Int ~([L(x,, . . . ,x,)) and Int Aut, [L(xl, . . . ,x,) in Section 2. In addition, the following notation will be used: Let T(x;, . . . ,x:), with IX:/ = ni and T(x;, . . . ,x:), with Ix;1 = ni, be copies of T = T(V). Then TuTuT

= T(xl,

. . . . x,,x;,

. . . ,x:,x;,

. . . ,x:‘).

3.8. Notation. Let 4(l,: II(V) + [L(V)u IL(V) be the comultiplication defined by &1j(u) = u + u’ for u E I/. Let 6:IL(V) + IL(V)~(L(V) be the homomorphism defined by 6(o) = u’ for u E V. Applying the universal enveloping algebra functor U, we obtain

M. Arkowitz, G. Lupton / Journal of Pure and Applied Algebra 102 (1995) 109-136

122

and a homomorphism U(S): {x1, . . . ,x,} for V we have

a comultiplication U(&,): T(V) + T(V)uT(V) T(V)+ T(V)u T(V). In terms of a basis U(4,i,)(xj) = xj + X>and U(6)(xj) = ~5. 3.9. Lemma.

Let

U(+,,,)(<) - 5-

tj E T(x,, U(d)(t)

is

. . . ,x,) in

be of length at Int T(xl, . . . ,x,,x;, . . . ,x:),

least then

two. 5 is

If in

Int T(xl, . . . ,xI). Proof. First we show that, for < of any length, if U(&,)([) - r E Int T(xl, . . . ,x,, x;, . . . ,x:), then 5 E Int 7(x1, . . . , x,). We proceed by induction over the length of 4. In the length 1 case, this is clearly true. Now suppose that the result is true for length m - 1 and suppose that 5 has length m. Write

5= 1

aiJXiXJ7

i, J

where .J = (j,, . . . ,jm_l) is a sequence of U($,r,)(C) - ‘E=

c”i,J{(xi

+

m -

&)U(~CI,)(XJ)

1 terms and -

XJ = Xj, ..* Xi,,_,.

Then

xixJ>

i,J

Hence, for each i, xJai, J(U($,,j)(XJ) - XJ) E Int T (XI, . . . , x,,x;, . . . ,x:). NOW the latter can be rewritten as U(c#+,,)(CJaiJxJ) - CJ a i J x J , so the inductive hypothesis implies that each ai, J E h. This completes the induction and proves the preliminary result. the lemma, suppose To prove that (U(&,)(5) - 5 - U@)(G) E Int T ,XL). Write r = xi J12iJXiXJ, SO that 1, . . . ,x*,x;, . . . (x U(4,1,)(5) - 5 - U@)(C) =

FxiT%J(U(+(I))(xJ)

Henceforeach i,~JaiJ(U(f$&(XJ) can be re-written U(t$+,,)(CJaiJXJ) preliminary result. 0

-

XJ)

-

+

Fx:T%J(“(4(l))(xJ)

XJ) E ht -

CJaiJXJ,

-

U(~)(XJ)).

T(xl, . . . ,X;, . . . ,X:). Again, this latter and the lemma follows from the

3.10. Proposition. Let #,1c/E Int %( T (x1, . . . ,x,)) and let 8 E Aut, T (x1, . . . ,x,). If O*C$= t,b, then 0oIntAut,T(xr, . . . ,x,). Proof. We use induction over the number of generators xi, with the inductive step proved by a secondary induction over the length. The main inductive hypothesis is that 0(xi)E Int T(xl, . . . ,x,) for all i c k. For i = 1 this is clearly true, since Qx,) = xi.

ht. Arkowitz, G. Lupton /Journal of Pure and Applied Algebra 102 (1995) 109-136

123

suppose that e(xi) E Int T (~1, . . . ,x,) for all i < k. Notice that this implies is also in Int T(xl, . . . ,xI) for all i < k. Write 0(x,)= + Qm(xk), where Qs(xk) is of length s. We will show that xk + Qdxk) + ... Qs(Xk) E hit T(Xl, . . . ,xr) for each s, by induction over s. Our secondary induction hypothesis is that Qs(xk) E Int T (x1, . . . ,x,) for each s I 1. This induction starts with 1 = 1 where the result is obvious. Now NOW

8-‘(xi)

bhxk)

= co* &(xk)

= (euo$e-‘(xk)

= (b+$(xk

- 8-‘&(Xk)

-

...

-

&‘Q,(Xk)).

Working up to congruence modulo terms of length 2 I + 2, and letting P denote the perturbation of 4 in a similar way to (l.l), we have $(xk)

= (f+o$(xk

- e-‘&(xk)

-

...

- m&k)

&(xk)

-

@-’

- Ql+dXk))

1+1

=

=

(cue) xk +

(

xk + xi

+

c

i

s=2

i

Qs(xk)

Qs(xk)

s=2

+ x: + u(a)(

i

s=2

-

~(h)@l+l(xk)) >

Q,cxk,)

s=2

1+1 +

vu

0)

1

Ps(xk)

-

be-

’

s=2

-

(~(h)(Q~+l(~k))

i

Qs(Xk)

s=2

-

Q~+lbk)

>

-

U(WQl+l(xk))).

Now II/and 4 are in Int %‘(T (x1, . . . ,x,)), and Qs(xk) E Int T (x1, . . . ,x,) for 1 I s I 1. Combining these facts with the remark above about 8-l, we see that U(&,)(Ql+l(xk)) - Ql+l(xk)) - U(h)(Ql+l(xk)) E Int T(xl, . . . ,x,,x;, . . . ,x:). Lemma 3.9 then gives that Q1+1(&) E Int T(xl, . . . , x,), which completes the secondary induction. Hence Qs(xk) E Int T (x1, . . . , x,) for all s, and the main inductive step is proved. This completes the induction and the result is proved. 0 For Lie algebra comultiplications, we do not prove as sharp a result as the above _ see Remarks 3.13. However the following proposition is sufficient for our applications.

let OEAut,L(xl, . . . ,x,). Zf Let $, II/E Int %(IL(xl, . . . ,x,)) d 8 * 4 = $, then SQs(Xj) E Int [L(xl, . . . ,xI), for all s and all j, where 8 is written e(xj) = xj + Qz(xj) + ... + Qm(xj) US in (1.2).

3.11. Proposition.

Proof. Let i: [L(xl , . . . ,x,) + T(xl , . . . , x,) be the inclusion into the universal enveloping algebra. Now U(0) E Aut, T(xl, . . . ,x,) and U(4), U(l(l) E Int %‘(T (x1, . . . ,x,)). Since e * C#J = * implies u(e) * U(4) = U(lc/),we apply Proposition 3.10 and conclude that U(d) E IntAut, T(xl, . . . ,x,). Then i(Qs(xj)) E Int T(xl, . . ,x,) for each s. According to [19, p. 2811 (see also [14, p. 169]), the linear mapp: T(V) + R(V) defined by

124

M. Arkowitz, G. Lupton /Journal of Pure and Applied Algebra IO.2 (I 995) 109-136

Ptxj, “* Xj,) = l/s Cxj,,C... 9CXj.-,~xj.l***11, is a left inverse for i. Hence Qs(xj) = pi(Q,(xj)), and since i(Qs(xj)) E Int T(xi, . . . , x,) we have for each generator xj and each S, sQs(xj) E Int R(x,, . . . ,x,). El Finally, we apply the preceding results to the map of orbit sets mentioned above. 3.12. Proposition. The map of orbit sets q:IntV(lL(x,,

. . . , x,))//IntAut,

Il(xi, . . . ,x,) + %‘(L)//Aut,L

induced by inclusion isjinite-to-one, where L = [L(xI, . . . ,x,).

Proof. We will write [4] for the equivalence class of a comultiplication in the orbit set Int%(lL(x,, . . . , x,))//Int Aut, L(xi , . . . , x,) and {b} for an equivalence class in the orbit set %‘(L)//Aut,L. Suppose that {4} is in the image of q with 4 E IntV([L(xl, . . . ,x,)). Let (0,) . . . ,&,} be the subset of the set of automorphisms 9 described in Definition 3.5, consisting of those 0, E 9 with = {[0,*4], . . . ,[&,*4]}. For f3j*C#JE IIlt%(L(X,, . . . ,x,)). We willshow that q-l(4) ,x,)) and that {4} = {$}. Then there exists 8 E Aut, L suppose that $ E IntV(L(xl, . . . 3.11 and Lemma 3.7, there exists with 8* 4 = II/. By Proposition 8~ IntAut,L(xi, . . . ,x,) such that &otlEF-. Now ~*I+G~Int%(L(xi, . . . ,x,)), and so 80 8 = ej for some j. Hence [$I = [g* $1 = [ej * 41 for some &.* = @ep++, j. q 3.13. Remark. Observe that Proposition Int%‘(T(xi, . . . , x,))//IntAut,

3.10 asserts that the natural map

T (x1, . . . ,x,) + U( T)//Aut,

T

is one-to-one, where T = T(xI, . . . , x,). Thus Proposition 3.12 is a Lie algebra analogue of Proposition 3.10, but with a weakened conclusion. In fact, a stronger formulation of Proposition 3.12 is true. This uses the following Lie algebra counterpart to Lemma 3.9: If < E [L(V ) is of length at least 3 and if 4(i)(l) - 4 - S(t) E Int [L(I’), then 5 E IntL( I’). The latter result can be proved with a rather delicate Hall basis argument. From this, one argues as in the proof of Proposition 3.10 and shows that in certain circumstances the map of orbit sets q is actually injective. For example, if each generator xi has odd degree, then q is injective. However, Proposition 3.12 suffices for our purposes. Combining Propositions 3.4 and 3.12, we have the following theorem. 3.14. Theorem. Let X be a finite cogroup and let Lx = E, (SZX) @ Q be the rational homotopy Lie algebra. Then the map of orbit sets

r: @:(-Wl~,(-9 -, ~(Lx)llAut, LX

M. Arkowitz, G. Lupton / Journal of Pure and Applied Algebra IO2 (1995) 109-136

induced by g :S’(X) + W(Lx) and h : b,(X)

+ Aut, Lx isjnite-to-one.

125

Consequently, if

%?(LX)//Aut, Lx is finite, then G?(X) is jinite. 3.15. Remark. The maps g:%‘(X) --) %(Lx) and h : C?(X)+ Aut, Lx also induce a map

of equivalence classes r”:@(X) + @(Lx). However, this map need not be finite-to-one, even when restricted to G?,(X). For instance, if X = S3 V S5, then G!?:,(X)has infinitely many elements (see Example 6.6( 1)). On the other hand, an easy computation shows that @:,(Lx) has exactly two elements (cf. Example 5.4(l)). Since we prefer to compute in the setting of the rational homotopy Lie algebra whenever possible, this explains why we focus on the the orbit sets %‘(X)//C~,(X) and %‘(Lx)//Aut, Lx rather than on the sets of equivalence classes g:(X) and @(Lx). In the following example we illustrate how, in applying the map Yof Theorem 3.14, the loss of torsion may lead to strictly fewer equivalence classes of comultiplications. In what follows, and also in the examples of Section 6, we adopt the following notation: If X is the wedge of spheres X = S”’ V a.. V S”‘, then rj E n.,(X) denotes inclusion into the jth summand. Similarly we use rj and rJ to denote those elements of rrn,,(X V X) given by inclusion into appropriate summands. The bracket in this context denotes Whitehead product. We will frequently use the fact that, if 8 E n&S’) is suspension, then left-additivity holds, i.e., for f, g E z,(X) we have (af+g)“B+B+go& 3.16. Example.

Let X = S’ V S19. Since X is of type (6,18), one sees easily that %Z(L,) is infinite. From [4, Theorem 3.141, however, it follows that Ws,(Lx) and hence %‘a(Lx)//Aut,Lx contains a single element. Since the map I: %(X)//&,(X) + g(Lx)//Aut, Lx restricts to a map I’: %?$C)//b,(X) + %‘=(Lx)//Aut, Lx, Theorem 3.14 now implies that %JX)//b,(X) is a finite set (cf. also Theorem 6.1 below). Indeed, the latter set contains exactly two elements, as we now show: According to [22, p. 1871, we have 7ri9(S7) = 0. It follows that b(X) z Zz x Z2 and g,(X) consists of the identity. On the other hand, it follows from Hilton’s theorem [ll, Theorem A] that a typical element of U(X) can be written c#+.,~,~), with &.,,Jzl) = z1 + i’i and

for integers a, b, c with 0 E 7r19(S’3) E Z2 denoting the generator [22, p. 1861. Since I3is a suspension, a may be reduced modulo 2. Now if c#+~,~,~) E %JX), then b = c = 0. This follows from [4, Theorem 6S.(ii)] and the well-known fact that, for q an odddimensional homotopy element, the triple Whitehead product [q, [QV]] is zero. Hence there are two elements in U,(X), represented by c&,,o,O)and &,o.O,. By the previous remarks 8*(X) is trivial, and so ‘Za(X)//&*(X) contains two equivalence classes. We remark that the calculation can be continued to show that b(X) acts trivially on Va(X), and so @,(S’ V S19) also has exactly two elements.

hf. Arkowitz, G. Lupton /Journal of Pure and Applied Algebra IO2 (1995) 109-136

126

4. Quadratic Lie algebra comultiplications The purpose of this section is to prove a result (Theorem 4.4) that essentially shows an associative comultiplication is determined up to isomorphism by its quadratic part. This result, which is of interest in its own right, will be used in Section 5. If C$ is a comultiplication of L = O_(xl, . . . ,x,) we write, as in (l.l), c#J(x~) = xi + xi + Es 2 zPs(Xi) where PJx,) is the perturbation term of length s. is the comultiplication 4.1. Definition. The t-fold part of C#I f#l(t)(Xi) = Xi +

Xi + C

&) defined by

ps(xi)-

s=2

Note that this depends on the choice of a basis x1, . . . , x, of L. The 2-fold part $C2Jof 4 is called the quadratic part of 4, and a comultiplication C$is called quadratic if C$= #C2,. This is consistent with the definition of 4(I) in Notation 3.8. As before let L u L u L = IL(xl, . . . , xr, xi, . . . , XL, x7, . . . ,x:‘). The following notation will be used extensively in this section. 4.2. Notation. Define homomorphisms

/?,y, 6: L u L + L u L u L by B (xi) = Xi + xi, in Notation 3.8 we Note that b = &,ul and y = lU&,.

P(x;) = x:l,r(Xi) = xi, Y(x;) = X: + x;; d(xi) = x:, 6(x:) = x:. AS

also regard 6: L+LuL.

4.3. Proposition. Let x E O_(xl, . . . ,x,, xi, . . . ,x:) be of length s 2 3. ZJ x + b(x) = y(x) + 6(x), then x = 4&t) - r - &{)for some l E [L(xl, . . . ,xI) oflength s. Proof. This is an immediate consequence of the proof of Theorem 3.11 in [4]. One replaces P,(Xj) in [4, 3.121 by x, and argues as in the rest of that proof. q We now state and prove the main result of this section. 4.4. Theorem. Zf 4 and II/ are associative comultiplications of the Lie algebra L = [L(Xl, . . . ,xr) such that 4~2)= $(z), then there exists tl E Aut,L such that

e**=4.

Proof. We construct 8 inductively over the generators Xi. We also use a secondary inductive argument. The primary inductive hypothesis is the following: There exists @km‘) E Aut, L such that (P- l) * Ic/)C2, = $C2j= 4(Z) and e(k-l) * II/ and 4 agree on all generators Xi with i < k. The induction begins by taking e(l) = 1, the identity automorphism. NOW suppose eCk-l) exists. We construct Ock)by a secondary induction over the bracket length. We write C$as in (1.1) and assume inductively that for some m 2 2,

M. Arkowitz,

G. Lupton 1Journal of Pure and Applied Algebra 102 (1995) 109-136

there exists q,,,, E Aut,L (a) q,,,) has the form

such that if i # k, if i = k,

xi

VCmj(xi) =

xk -

127

c,“=,&

where each r, is of length s and (b) if R is the perturbation (qm) * (Yk- I) * $)txk)

=

of the comultiplication

xk

+

x6

+

c

Q,,) * 13’~~‘) * Ic/,namely, (4.7)

R&k),

s>2

then P&k) = R&k) for S = 2, . . . ,m, where P is the perturbation Claim 1. Pm+1(Xk) =

Rrn+dXk)

+

&&rn+d

-

trn+l

-

&trn+d

of 4. for

some

tm+l

E

m + 1.

rL(Xl, *..,x,)oflength

Proof of Claim 1. Since 4 is associative, (4 u l)$ = (1 IJ 4)4, we have

sF2ps(xk)+(6u1) ,F2’s’ (

k)

=tl

x

$1

>

,F2c’

u

k)

(

+a

x

c

)

ps(

L2

k) x

)-

t4.@

Similarly, since II/, and hence qrnj * 0 ck- ‘) * Ic/,is associative, (4.7) yields sF2s(xk)

+ ($” ‘1( SF2 Rs(x,k) = c1u(6) sF2M x)k)

+

6

(

,F2

R,( k) . (4.9)

x)

We subtract (4.9) from (4.8) and obtain 1

tps(xk)

-

R&k))

+

(4

u

1)

1 ( SZltl+l

szm+1

=

(l”d))

1

tp&k)

-

R&k))

tpstxk)

R&k)) >

+

6

c ( s>m+l

1

( s>m+l

-

tpstxk)

-

Rsbk))

.

(4.10)

>

We extract the length m + 1 terms from (4.10) and get Pm+lbk) =

- &+l(Xk)

dPm+l(xk)

-

+

fi(Pm+l(Xk)

&n+,(xk))

-&+l(xk))

+

~((pm+l(xk)

Claim 1 now follows from Proposition 4.3. Proof of Theorem 4.4 (continued). Ylm+l(Xi) =

i

&+l(xk))

0

With &,,+1 as in Claim 1, define Q,,+1 E Aut, L by

if i # k, if i=k.

xi &-&,,+1

We let T be the perturbation hn+1 * (qtrnj* e’k- 1) *

-

$))(xk)

of the comultiplication =

xk

+

x;

+

c s>2

T&k).

Q,,+1 *(q,) * Btk- ‘) * Il/), so

128

M. Arkowitz, G. LuptonlJournal

of Pure and Applied Algebra 102 (1995) 109-136

Claim 2. For s = 2, . . . ,m + 1, T&Q) = Ps(xrJ. Proof of Claim 2. We have (Vm+l*tl~,)*~(k-l) * Ic/)bk)

=

(?m+l

U?m+l)((%n,

* 8 (k-

$)bk)

“*

+

~(&n+

1))~

(4.11) since for generators xj with j < k, (q,,,) * 4 (k-1)* Ic/)(xj) all the terms of length I m + 1 from (4.11) and get xk

+

x;

=

h!+1

+

1 s=2

xk

u%n+l)

-

* $)(Xj)

= +(xj).

Take

Tshk)

xk

+

x;

+

i

‘tin+

1 +

xi

p&k)

+

&+l(Xk)

+

&l,(&n+l)

s=2

(

=

= (es

-

~(‘&I+

1)

+

>

5

ps(xk)

+

&n+

lbk)

+

&l,(tm+

1).

s=2

Thus, III+1 c

T&k)

=

.9=2

f

p&xl,)

+

&,+l(Xk)

+

&l&,+1)

s=2

-

tit+1

-

s(&n+l)

=

m~‘ps(xk). s=2

This proves Claim 2.

17

Proof of Theorem 4.4 (continued). By construction, rf,,,+1 0q,) has the form (&?I+1 ’ V(m))(xi)

=

xi xk

-

cs”=‘3’

5,

if i # k, if i = k.

We set q(,+ i) = qm+1 0 q(,) to complete the inductive step of the secondary induction. We start the secondary induction by setting u(2)equal to the identity homomorphism and note that v(2) satisfies the desired condition since (Otk)* $)(2i = &2). Therefore there exists r~E Aut, lL(xl , . . . ,x,) such that tJ(Xi) =

Xi xk--~s>3&

if i # k, if i=k,

(4.12)

where 5, is of length s and q * Ock-‘) * $ and $J agree on all generators Xi with i I k. It follows from (4.12) that (q * B’k- ‘) * +)(2, = (O(k-1)* $)(2, and thus equals &2, by the primary inductive hypothesis. By setting 0(k) = q 0 Otk- ‘), we complete the inductive step. Finally, the theorem is proved by putting 8 = r3(‘),since r is the number of generators of L = U-(x1, . . . ,x,). 0 4.5. Remark. There is an analogue of Theorem 4.4 for tensor algebras as follows: If 4 and $ are associative comultiplications of the tensor algebra T = T(xl , . . . , x,) such that $(2) = tic2), then there exists 8 E Aut, T such that 8 * $ = 4. The proof is obtained by carrying over the proof of Theorem 4.4 mutatis mutandis to tensor algebras. We apply this tensor algebra result in [S].

M. Arkowitz, G. Lupton 1Journal of Pure and Applied Algebra 102 (1995) 109-136

129

We finish this section with an example that illustrates Theorem 4.4 cannot be improved. If an associative comultiplication C$E %Za(L)has quadratic part 4(Z) which itself is an associative comultiplication, then Theorem 4.4 implies that C#J is equivalent to its own quadratic part &. Our example shows that in general this situation need not pertain. Let L be the Lie algebra IL&,, yzp, z 3p) where subscripts denote degrees, and assume that 1xp 1= p is even. Let C$be any comultiplication of L with perturbation P and write P(x) = 0, and

P(Y)

= 4x,

P(z) = b[x,y’]

x’l

+ c[x’,y] + d[x,[x,x’]]

+ e[x’,[x,x’]]

forsomeabcdeEQ.If6EAutL,then 9 999 e(x) = ax,

8(y) = /?y and

6(z) = yz + S[x, y]

for a, /I, y, 6 E Q with a, /I, y non-zero. 4.6. Lemma. With L, C$and 9 as above: (1) 4 is associativeoab + e = 2d and ab + ac = d + e. (2) (a) (0 * 4)(x) = x + x’, (b) (e * dw

= y + Y’ +

$Cx,x~l,

+ da’/3 - aa BY

Proof. The proof is a straightforward

CT Cx,x’ll

+ ecr3pgyna’6 [x’, [x,x’]].

but long calculation, and hence omitted.

0

4.7. Example. Define a comultiplication

C#Jby taking a = 1, b = 2, c = - 1, d = 1, e = 0 in the above. Then 4 is associative, by Lemma 4.6, but it cannot be equivalent to any quadratic comultiplication. To see this, suppose that 8 * 4 is quadratic, where 8 E Aut L is as described above. Then Lemma 4.6 implies a38 - a26 = 0 and a26 = 0. From the latter, 6 = 0. This implies a”fl = 0 which is impossible. Notice, in particular, that 4 cannot be equivalent to its own quadratic part. Indeed, it follows from Lemma 4.6 that $C2,is not associative.

5. Lie algebra comultiplications In this section we consider comultiplications of Lie algebras and prove that certain orbit sets are finite (Proposition 5.3), by using Theorem 4.4 on the quadratic part of comultiplications. This result together with Theorem 3.14 will yield our main theorem in Section 6 (Theorem 6.1) on the orbit set of comultiplications of a finite cogroup.

hf. Arkowitz, G. Lupton /Journal of Pure and Applied Algebra 102 (1995) 109-136

130

We adopt the following standard convention for taking a sum over an index set: Whenever the index set is empty, the sum is defined to be zero. 5.1. Proposition. Let r#~be a comultiplication in W&(x, P such that

Pz(Xi)=

1 aj[Xj,X~l + C aj,k([Xj,XLl

j, npdd

. . . ,x,)) with perturbation

+ CX;,Xkl)

j
a:, ai,k E Q. Then q5 is equivalent, by an element in Aut, [L(xI, . . . ,x,), to the comultiplication c$(~,.

for

Proof. Define 0 E Aut, lL(xr, . . . ,x,) by 0(x<) = xi +

1

$

[xj,xj]

-

1

af,k

[xj,&].

j
j, n,odd

We determine (0 * 4)(2,. Observe that modulo terms of length 2 3, OF’(Xi)

E

Xi

-

j,nzdd$[Xj,

Xj]

-

C

Uj,k

[Xj,

Xk].

j
Therefore, working up to congruence modulo terms of length 2 3, we have

te*4)txi)

E

c teue, x~+ xi + j,ajodd

aj[xj,

xi] + 1

j
_i,zdd:[ j+ + - 1 X

X>,Xj

x;]

aj,k(

Uj,k[Xj

[xj,

+

xi] + [xi, xk])

xi,xk

+ XL]

.

jik

Hence (0 * $)(z)(xi) = xi + xi. Since 0 * $J and &r) are associative, it follows from Theorem 4.4 that there exists 6’ E Aut, lL(xr , . . . ,x,) such that 8’ * (0 * 4) = 4(r). Thus 4 is equivalent to 4(l) via 8’8 E Aut, R(xr, . . . ,xr). 0 Recall that Va,(L) 5 Va(L) consists of the Lie algebra comultiplications are associative and commutative.

of L which

5.2. Corollary. Let L = (L(xI, . . . ,x,) with lxil = ni. Suppose that, for each i, (a) ni # nj + njfor every j, k with j < k and (b) Iti # 2nj for every j with nj even. Then the map of orbit sets 1: %,,JL)//Aut, L -+ %‘JL)//Aut* L induced by inclusion is a bzjection. Proof. Clearly Iz is one-one. Now let 4 be any associative comultiplication of L. Under our assumptions, 4 satisfies the hypotheses of Proposition 5.1 with the aj,k all zero. Thus 4 is equivalent to the associative and commutative comultiplication h.

0

M. Arkowitz, G. LuptonlJoumal

of Pure and Applied Algebra IO2 (1995) 109-136

131

5.3. Proposition. Let L = [L(XI, . . . ,X,) With [Xii = %. (1) Iffb each i, (a) ni # nj + Q for every j, k with j < k and (b) ni # 2nj for every j with nj even, then the orbit set wa(L)//Aut,L contains a single element. (2) The orbit set %=,(L)//Aut, L contains a single element. Proof. By Corollary 5.2, it suffices to show (2). Let 4 E %J[L(x~, . . . ,x,)) have pertubation P. Since 4 is commutative, TPz(xi) = Pz(xi), where T is the switching homomorphism. From this it follows that #J satisfies the hypotheses of Proposition 5.1. Thus 4 is equivalent by an element of Aut, [L(xI, . . . ,x,) to the comultiplication &,. 0 5.4. Remarks. (1) The conclusion of Proposition 5.3(l) need not be true without the hypothesis on ni. For example, if we set L = [L(x,, y2,,), where subscripts denote degree and n is even, then Aut, L consists of the identity automorphism. But for every m, the comultiplication 4 defined by 4 (y) = y + y’ + m [x, x’] is in g,(L). Thus the orbit set %?JL)//Aut, L is infinite. (2) Under hypothesis (1) of Proposition 5.3, every associative comultiplication on L is equivalent to &). Thus if 4 is an associative comultiplication of a Lie algebra L which satisfies (l), then L is ‘primitively generated’ with respect to 4. The following is an immediate consequence of Proposition 5.3. 5.5. Corollary (cf. [4, Theorem 3.181 and [9, p. 83). ZfL = U-(x,, . . . ,x,) with [Xii odd for every i, then %Za(L)//Aut,L consists of a single element. Thus any two associative comultiplications of L are equivalent via an automorphism in Aut, L. 0 5.6. Remarks. Corollary 5.5 and part (2) of Proposition 5.3 are duals of well-known results about diagonals on commutative graded algebras. Corollary 5.5 is dual to the Leray-Samelson theorem [17, 7.201, which states that a commutative Hopf algebra with generators in odd degrees is primitively generated. Proposition 5.3(2) is dual to the result of Milnor-Moore [ 17, 4.181 that a commutative Hopf algebra with a commutative diagonal is primitively generated.

6. Comultiplications of a finite cogroup We recall from Section 2 that a finite co-H-space X has the rational homotopy type of a wedge of spheres, XQ G Sz” V ... V Sz+’ for integers nl I 1.. I n,, and that (n 1, *-* 9n,) is the type of X. The following is the main result of this section.

M. Arkowitz, G. Lupton JJournal of Pure and Applied Algebra 102 (1995) 109-136

132

6.1. Theorem. Let X be ajinite cogroup oftype(q, . . . , n,). (1) Iffir CY.ZC~i, (a) ni # nj + nkfor euery j, k with j < k and (b) ni # 2njfor every j with nj even, then the set of orbits Wa(X)//b,(X) isBnite. Consequently G?,(X) is$nite. (2) The set of orbits %gac(X)//S,(X) is Jinite. Consequently G!?,,(X) is Jinite.

Proof. (1) Let Lx = 7~,(s2X)@ Q = [L(q, . . . , x,) with [xi1 = ni. Consider the commutative diagram

W-W/&‘,(X) L

WLx)llAut,Lx

where r is as in Theorem 3.14, the vertical maps are induced by inclusion and r’ is the restriction of r. Since r is a finite-to-one map by Theorem 3.14, r’ is a finite-to-one map. By Proposition 5.3, %=(Lx)//Aut*Lx consists of a single element. Thus %‘,(X)//&,(X) is finite. The second assertion of (1) follows from the first since the map qa(X)//S,(X) -t %JX)//&‘(X) = @JX) induced by the inclusion 8*(X) E b(X) is onto. Part (2) is proved analogously to part (1). 0 6.2. Corollary. ZfX is aJinite cogroup of type (q, . . . ,n,) with all ni odd, then ga(X) is jinite.

6.3. Remark. In [9, Theorem I], Curjel proves that there are finitely many equivalence classes of homotopy-associative multiplications of a finite H-space. We regard Corollary 6.2 as dual to Curjel’s theorem, and justify this in the following way. A finite H-space has an oddly generated rational cohomology algebra. A finite co-H-space, on the other hand, has an oddly generated rational homotopy Lie algebra if and only if it has type (ni , . . . , n,) with all ni odd. 6.4. Remark. Clearly Theorem 6.1 holds with the set of homotopy classes of comultiplications of X whose rationalization is homotopy-associative replacing the set of homotopy-associative comultiplications of X. A similar remark holds for comultiplications which are rationally both homotopy-associative and homotopy-commutative. We illustrate Theorem 6.1 with several examples. The strength of our results becomes apparent when the ad hoc approach of Example 3.16 is taken instead, as in the following example. 6.5. Example. Let Y = S3 V S4 V S*. Theorem 6.1 implies G?,(Y) is finite, since Y is of type (2, 3, 7). Alternatively, as in Example 3.16 use Hilton’s theorem to write a typicalelement of V(Y) as 41 for I = (A,, &, A3, &, &, &, &, la, &, ;liO, ;lil, &).

M. Arkowitz, G. Lupton 1Journal of Pure and Applied Algebra 102 (I 995) 109-l 36

133

with 4~(11) = z1 + z;, 41(z2) = l2 + I;, and

In this last equation a, /_Iand y denote generators, respectively, of na(S5) g Z24, ns(S6) g Z2 and rra(S’) z Z2 [22, p. 1863, A1is reduced modulo 24 and A2, &, A4, A5 and 16 are reduced modulo 2. We use the fact that all the generators are suspensions to describe +1(z3) in this way. Now consider the restrictions on the lj in order that 41 be homotopy-associative: By a straightforward argument using commutativity and the Jacobi identity for the Whitehead product as in [4, Example 6.91, we see that A5 = 126= 0. By applying [4, Theorem 6S(ii)], in conjunction with the result mentioned in Remark 3.13, it is possible to show that llj = A, forj = 8,9, 10, 11 and 12. It now follows that we can write

+ w-h

+ t;,cz1 + il,l2+

&!I1- CE1,[21,Z2]] - [E;,[I;,I;]]).

Furthermore, a straightforward computation shows that any comultiplication of this form is homotopy-associative, so %a(Y) contains infinitely many elements. Proceeding as in Theorem 4.4, we now observe that if 4I is as in the last expressions, and if t?~&‘*(Yf is given by 8(r1) = r1,8(z2) = z2 and @(z,)= z3 + A7[rlr[r1,z2]], then 6* A = &A, ,A,,~,.~,,o,o,o,o,o,o~o,o~. Hence +3dY) isfinite. In Example 3.16 we had wa(X) finite with b(X) acting trivially. In this example, however, %?a(Y) is infinite and it is necessary to take into account the action of b( Y ). To do so, we basically mimic the proofs of results such as Theorem 4.4. This example suggests that any argument to show ga(Y) is finite, which is to be generally applicable, will in one form or another use some of the key ingredients of Theorem 6.1. From the previous working, if Y = S3 V S4 V S*, then ga(Y) contains at most 24 x 23 elements. It is natural to ask whether we can be more precise. Our calculations may be continued along similar lines to show that ca(Y) contains at most 24 x 22 = 96 elements. Standard methods such as Ganea’s result [lo, Corollary 3.51 do not apply here. It would be interesting to know the exact number of elements in @&S” v s4 v S8). Our next examples show the conclusion of Theorem 6.1(l) may not hold, if the hypothesis is relaxed. We will see that even when X is a wedge of two or three spheres, there are many interesting possibilities, and our examples will be of this type. First note that the hypothesis of Theorem 6.1(l) may fail to hold in one of two ways: either ni = 2nj with nj even or ni = nj + nk with j < k.

134

M. Arkowitz, G. Lupton / Journal of Pure and Applied Algebra 102 (1995) 109-136

6.6. Examples. (1) Let X = S”+’ V S”‘+l, where n is even. We show that G?‘,(X)is infinite. By [4, Theorem 6.61, wa(X) is infinite. On the other hand, 8’(X) is a finite group (see, for example, Proposition A.2 below). Therefore the orbit set g:,(X) is infinite. (2) Let X = S”+’ V S”+’ V Sm+“+l with m < n and n # 2m. We show 5?,(X) is infinite. Define for every pair of integers s, t a homotopy-associative comultiplication

= 12 + 1; and &,&) = r3 + 1; + 4 (S,f) of X by &,&i) = i1 + G,&,&) implies 1s - tl = 1u - VI. Assuming this sClr,&l + tCli,l21. We claim 4~) - 4~ claim, we easily see that ‘G??,(X)is infinite since, for example, the comultiplications &Oj represent distinct equivalence classes for each s. We now prove the claim: Suppose f E d(X) Then f has the form and f* &S,fJ= &,vj. f(zi) = pi1,f(r2) = qr2 + tl andf(r,) = rz3 + k[z1,r2] + /I, where each of p, q and r is either 1 or - 1, k is an integer and u E rr,,+i(X) and /I E rc,,,+“+i(X) are torsion elements. Now a straightforward calculation, working up to congruence modulo torsion, yields u - v = pqr-‘(s - t). Finally we show that, although g:,,(X) is always finite, the set of equivalence classes of homotopy-commutative comultiplications g,(X) could be infinite. 6.7. Example. Let X = S”+l V S*‘+l, r 2 3. Then by [4, p. 103 and Proposition 4.21, Vc(X) is infinite. However, b(X) is finite and thus g:,(X) is infinite.

Appendix. The group of self-homotopy on homology

equivalences

that induce the identity

Let Y be a l-connected, finite complex. In this appendix we prove that a,( Y ) and all of its subgroups are finitely generated. An analogous result for homotopy groups appears in [2]. Our argument depends heavily on the work of Maruyama. We use the notation of [16]. For n 2 3, let V = Sl-’ V ... V S:-’ be a wedge of Y(n - 1)-spheres, A a l-connected, finite complex of dimension I n - 1, f: V + A an arbitrary map and X the mapping cone off: Then there is a cofibre sequence

where i is the inclusion. Assume VI = Sl-’ V ... V Sz-;’ E V satisfies kerf, = H,_,(V/V,) = H,_,(S;;;,‘, v ... V S:-‘), where f,: H,_,(V)+H,_,(A) is the induced homomorphism. Denote by G(V) the subgroup of b(V) consisting of all elements of the form 1 + y for y E [V,, V/V,]. Define a subgroup 8,(A) of B,(A) by 8,(A) = {gEb*(A)lgf=f8

for some deG(V)).

Consider the map i,: [A, A] + [,4,X]

induced by i and the set i,B,(A)

c [A,X].

M. Arkowitz, G. Lupton / Journal of Pure and Applied Algebra IO2 (1995) 109-136

A.l. Lemma.

135

The set i, 8’,(A) can be given group structure so that i, : if_*(A) + i, I, (A)

is an epimorphism. Proof. For g, h E a,(A), define i,(g)* i,(h) = i,(gh). We show that this is well-defined. Suppose i,(g) = i, (g’) and i,(h) = i,(h’). Then igj = $5 for some 6 E G( V ). But i$Sis the constant map and so ig induces a map& X +X such that gi = ig. Thus i,(gh)

= gih = @h’ = igh’ = ig’h’ = i,(g’h’). This shows that the multiplication &Z-,(A) is well-defined. The rest of Lemma A.1 now follows easily. 0

in

Next consider the subgroup i,[CV, A] of [CV, X]. A.2. Proposition. &a,(A)

There are homomorphisms

such that the following

i,[CV,

A]L

R,(X)2

sequence

,I: i, [CV, A] -+ b,(X)

and i,: a,(X)

of groups and homomorphisms

-+

is exact

&a,(A).

Proof. This is proved in [16, Theorem 1.33.

0

We say that a group G satisfies the maximal condition if G and all of its subgroups are finitely generated. A.3. Proposition. If b,(A) maximal condition.

satisftes

the maximal

conditions,

then a,(X)

satisjies

the

Proof. Since i,: [CV, A] + [CL’, X] is a homomorphism of finitely generated abelian groups, i, [CV, A] satsfies the maximal condition. It suffices by Proposition A.2 to show that &$‘,(A) satisfies the maximal condition. Since b,(A) satisfies the maximal condition, so does 8’-,(A) G 8, (A). However, i, : g,(A) -+ i, 8, (A) is an epimorphism, and so &r!?,(A) satisfies the maximal condition. 0 A.4 Proposition. Zf Y is a 1-connected,jnite complex, then L?*(Y) satisjies the maximal condition. In particular, 8, (Y ) is jinitely generated. Proof. We sketch the proof which proceeds inductively over the skeleta Y” of Y. For

the inductive step from Y n- ’ to Y ” we apply Proposition A.3 with A = Y n- ‘, A the map which attaches n-cells to A to form Y” and X = Y”. The existence of VI E V such that kerf, = H,-,(S~~;,‘, V ... V S:-‘) is assured by [16, Lemma 2.11. This completes the sketch of the proof. 0

f: I/= s;-’ lJ ... v s;-’ +

Acknowledgement

We are grateful to Hans Scheerer for an argument Lemma 2.2.

that led to the proof of

136

M. Arkowitz, G. Lupton f Journal of Pure and Applied Algebra IO2 (1995) 109-136

References [1] M. Arkowitz, and C. Curjel, On the number of multiplications of an H-space, Topology 2 (1961) 205-208. [2] M. Arkowitz and C. Curjel, The group of homotopy equivalences of a space, Bull. Amer. Math. Sot. 70 (1964) 293-296. [3] M. Arkowitz and C. Curjel, Groups of homotopy classes, Lecture Notes in Math., Vol.4 (Springer, Berlin, 1964). [4] M. Arkowitz and G. Lupton, Rational co-H-spaces, Comment. Math. Helv. 66 (1991) 79-108. [S] M. Arkowitz and G. Lupton, Minimal models of loop spaces and suspensions, Manuscr. Math. 79 (1993) 415-433. [6] I. Berstein, Homotopy mod C of spaces of category 2, Comment. Math. Helv. 35 (1961) 9-14. [7] I. Berstein, On cogroups in the category of graded algebras, Trans. Amer. Math. Sot. 115 (1965) 257-269. [8] A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math., Vol. 304 (Springer, Berlin, 1972). [9] C. Curjel, On the H-structures of finite complexes, Comment. Math. Helv. 43 (1968) l-17. [lo] T. Ganea, Cogroups and suspensions, Invent. Math. 9 (1970) 185-197. [1 l] P.J. Hilton, On the homotopy groups of the union of spheres, J. Lond. Math. Sot. 30 (1955) 154-172. [12] P.J. Hilton, G. Mislin and J. Roitberg, Localization of Nilpotent Groups and Spaces, Notas de Mathemitica (North-Holland, Amsterdam, 1975). Cl33 P.J. Hilton, G. Mislin and J. Roitberg, On co-H-spaces, Comment. Math. Helv. 53 (1978) l-14. [14] N. Jacobson, Lie Algebras (Dover, New York, 1979). [15] A.G. Kurosh, The Theory of Groups, Vol. II (Chelsea, New York, 1956). [16] K.-I. Maruyama, Localization of self-homotopy equivalences inducing the identity on homology, Math. Proc. Cambridge Philos. Sot. 108 (1990) 291-297. [17] J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. Math. 81 (1965) 21 l-264. [18] J. Neisendorfer and T.J. Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978) 565-580. [19] D. Quillen, Rational homotopy theory, Ann. Math. 90 (1969) 205-295. [20] J.-P. Serre, Lie Algebras and Lie Groups (Benjamin, New York, 1965). [21] D. Tan& Homotopie Rationelle: Moddles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics, Vol. 1025 (Springer, Berlin, 1983). [22] H. Toda, Composition Methods in the Homotopy Groups of Spheres, Annals of Math. Studies, Vol. 49 (Princeton Univ. Press, Princeton, 1962). [23] G.W. Whitehead, Elements of Homotopy Theory, GTM 61 (Springer, Berlin 1978).